ON THE MOTION OF PENDULUMS. 49
the last figure being in each case in the 12th place of decimals. We thus get
9r-*r'(i)=- 1-9635102, A = + '5772158* ............ (106).
34. When m is large, it will be more convenient to employ series according to descending powers of a. Observing that the general term of FB (a) as given by (88), in which Dr = 0, is
wo get for the general term of F^ (a)
M r, ___ [1 . 3. - .(2. - 3)]= f (2t- 1)2 _ 2t-l] *- ; 2.'4.".(2*-2y(4«a)pio*r 2t.*«* 2a }'
and the expression within brackets is equivalent to
whence
( 1 % V2 3 5
dW '((A — fyV"m" i• j__1 __ ' • •*••**• «'
and wo find by actual division
85. When many terms are required, the calculation of the \ coefficients may be facilitated in the following manner.
j Assuming afr^ (a) = v (<t) Al, (M), we have
*v (") -K' ff oo -«" ^ («) + «"* (woaj/y; w-
Substituting in the difle.re.ntial (ujuation (<S5) which I<\ lias to satisfy, we p;et
f/.i/' (a) + [v (a)}'2 - uM = 0...............(107).
Assuming
/ \ iiit/' \~1' _i -i f N""* _L ^ i ^^^
* (A is in Hid the well-known transcendent called Hitler's Otiwt.itnt., tho value of which is -r>77'21;*><>(;•{<) it<«. This, whic.li I oiitfht to lia,v<5 known, was iiointe.d out to me just after the puhiieation of the paper by my friend Prof. V. Newman.]
S. III. 4